Out-of-plane stability of gusset plates using a

simplified notional load yield line method

2017 NZSEE Conference

B. Zaboli & G.C. Clifton

Department of Civil and Environmental Engineering, University of Auckland, Auckland.

K. Cowie

Steel Construction New Zealand, Auckland.

ABSTRACT: Gusset plates are a key component of braced frame systems, connecting the

braces to the framing system. With traditional concentrically braced frames (CBFs), the

braces are designed for controlled inelastic action, which involves brace buckling when the

brace is in compression. To avoid the undesirable effects of brace buckling, buckling

restrained braces (BRBs) have been developed. BRBs allow a brace to yield in compression

without global buckling, thus making the brace of similar stiffness and strength in tension

and compression. Testing on individual braces has demonstrated that BRBs can perform

very well, however the brace can also fail prematurely if its connections are not

appropriately designed and buckle before the brace core yields in compression. Despite the

importance of gusset plates, their behaviour has not been well researched, with engineers

using a design method originally proposed by Thornton (1984). This is a column analogy

used to describe plate behaviour and a number of recent studies have shown that while this

method is too conservative in CBF connections, is not reliable in buckling restrained brace

frame (BRBF) connections. In this paper, a simplified notional load yield line model is

proposed for both CBF and BRB systems, which can adequately take into account the actual

collapse mechanisms of brace to frame connections, ensuring gusset plate stability is

maintained as required in each system. A comparison of several experimental test results

and those of the proposed method is presented showing the method is suitably conservative

for application to both CBFs and BRBFs.

1 INTRODUCTION

1.1 Background

Historically, CBFs have been used to resist lateral earthquake loads in steel frame systems. CBFs provide

excellent lateral stiffness to prevent damaging story drifts in small to moderate earthquakes, but are not

as effective in severe earthquakes due to the system degradation in strength and stiffness from brace

bucking in compression. Recent research has led to the development of BRBs to overcome the problems

related to buckling of bracing elements. BRBs can dissipate significant hysteretic energy during both

tension and compression cycles and allow the building to maintain similar stiffness and develop similar

ductility in both directions of cyclic action along a given principal axis.

Gusset plate connections are widely used to connect the diagonals to the framing systems. Due to the

complexity of stress distribution, boundary conditions, and geometry in different gusset plate

configurations, the stress analysis in the gusset plates is difficult and their actual compressive behaviour

is not well known nor easily determined. For design purposes, Thornton proposed a column-based

method for predicting the buckling capacity of the gusset plate according to the early research conducted

by Whitmore (1952). Gusset plates were considered as rectangular columns with an effective width

determined by a 30° stress trajectory (Whitmore width), and the Euler buckling capacity was determined

using an effective length factor (K-factor) of 0.65 and the average buckling length (Figure 1).

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Figure 1. Geometry of Thornton equivalent column (Whitmore 1952).

In the most CBF connections, the Thornton method is conservative because it does not consider the plate

action and the out-of-plane restraint provided by the plate outside the boundaries of the effective width.

Gross (1990) carried out full-scale tests on braced frame subassemblages and noted that the Thornton

method, even with a K-factor of 0.5, could considerably underestimate the actual buckling capacity of

the gusset plates. Higgins et al. (2013) tested six large-scale gusset plate specimens representing the I-

35W Mississippi River Bridge, which collapsed due to buckled gusset plates, and indicated that the

largest experimental K-factor was 0.8 and K-factors less than unity can be applied to estimate the

compressive capacity of gusset plates that buckle in a sway mode. As an important observation, all six

specimens failed by sway buckling mode under the yield-line (Figure 2(a)). Yam & Cheng (2002)

reported thirteen full-scale tests, and it was also deduced that, owing to stress redistribution in corner

gusset plates, using a stress trajectory angle of 30° could be too conservative and a 45° dispersion angle

would give a more reasonable prediction of buckling capacity. A more comprehensive approach was

suggested by Dowswell (2014) in which a variable stress dispersion angle was formulated to estimate

buckling capacity of gusset plates with different configurations and inelastic deformation capacity.

Recent research conducted by Dowswell (2016) developed a notional load yield-line (NLYL) model for

gusset plate stability that effectively considers many critical parameters that control the compressive

behaviour of gusset plates, such as the initial imperfection, yielding of the section, reduction in stiffness,

large eccentricity, second-order effect, and plate action. However, this method utilizes a yield-line

pattern and buckling length that is not consistent with the experimental observations.

On the other hand, BRBF connections have been identified as a critical area where insufficient research

has been undertaken. Currently, engineers employ a method based on the CBF connection behaviour.

However, BRBs have different structural characteristics to normal braces which may lead to an

instability failure of gusset plates if designed to a method based on CBF connection behaviour. Tsai &

Hsiao (2008) and Chou & Liu (2012) tested full-scale BRBFs and reported the out-of-plane buckling of

the gusset plate. Consequently, they recommended that the K-factor should be 0.65 and 2, for the cases

stiffened and regular gusset plate respectively. However, as can be inferred from Figure 2(b), the tests

exhibited a plastic failure mode over the yield-line. Therefore, an increased K-factor would not be the

best approach to cover this failure mode.

Figure 2. Two different gusset plate failure modes: (a) gusset buckling mode under the yield-line (Higgins et al. 2013); (b) plastic failure mode over the yield-line (Tsai et al. 2008).

In a pioneering study undertaken by Takeuchi et al. (2014), focus was given to the moment transfer

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capacity of the restrainer end, and a set of sophisticated equations for BRB stability was developed.

Although this method considered many details of the BRB anatomy as well as the rotational stiffness of

the gusset plate, in practice obtaining the rotational stiffness of different gusset plate configurations is

far from straightforward. It also did not take account of the gusset buckling mode under the yield-line.

1.2 Current design practice

The most recent American guidelines such as the AISC Seismic Design Manual (AISC 2012), and the

AISC 29 Steel Design Guide (Muir & Thornton 2014) have adopted the Thornton method and by using

the AISC column curve, the compressive strength, initial imperfection of 𝐿 1500⁄ , and accidental

eccentric loading are considered. The current recommended values for the K-factor are 0.5 for a corner

gusset plate, as established by Gross (1990), and 1.2 for a single brace gusset plate that is connected at

only one edge. In all cases, the middle length to the nearest support point (𝐿1) is used as the buckling

length. As an alternative, these guidelines reference the K-factors and buckling lengths suggested by

Dowswell (2006, 2012) that were calibrated for different gusset configurations to provide more accurate

solutions. Similarly, in New Zealand codes of practice, the Thornton method has been applied along

with the NZS 3404 (1997) column curve. HERA R4-80 (Clifton 1994) followed the K-factor of 0.5 and

the average buckling length by Gross (1990), however, in NZS 3404 the K-factor was changed to 0.7

and applied the clear buckling length that is the middle length.

1.3 Research objectives

In view of the above, it can be seen that there is no comprehensive and robust design procedure for

gusset plates available, to capture very well the key parameters governing the stability of the systems.

This is especially true in BRBFs. The main objective of this study is to develop a practical and more

logical design procedure, which can adequately take into account the actual collapse mechanisms of the

gusset plates in both CBF and BRBF connections using a simplified notional load yield line model.

2 PROPOSED DESIGN PROCEDURE

As a general stability rule, larger initial imperfection means more lateral load and less stability. Thus,

the accuracy of a gusset plate buckling capacity approximation would depend on the level of out-of-

plane initial imperfection and deformation. The initial imperfection as well as deflection on the gusset

that is triggered by the brace end can induce bending moment on the folding line of the gusset and

subsequently plastify it when enough strength is not provided. Creation of this hinge line will

significantly reduce the elastic buckling capacity of the gusset plate and as a result it may buckle even

though it was designed to remain elastic under the prescribed axial load using an effective length

calculation method. For this reason, the starting point has to have a reasonable estimate of the in situ

out-of-plane imperfection, and then the notional load at the brace end becomes:

𝑁 = 𝜃𝑖𝑁⋆ (1)

𝑁 = 𝜃𝑖𝑁𝑐𝑢⋆ (2)

Where 𝑁 = notional load at brace end on the yield-line, 𝜃𝑖 =total initial imperfection angle,

𝑁⋆ = overstrength compression capacity of the brace in CBF, 𝑁𝑐𝑢⋆ = overstrength compression capacity

of the core in BRB.

2.1 Initial imperfection

In CBFs the initial imperfection is a combination of out-of-plumbness of the brace and out-of-flatness

of the gusset plate with the recommended values (Dowswell 2016) of 1 500⁄ and 1 100⁄ respectively

(Figure 3(a)). In addition, to consider the reduced stiffness of the gusset plate due to the out-of-plane

displacement, the out-of-flatness of the gusset plate is magnified by a factor of two (Dowswell 2016):

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Figure 3. (a) Initial imperfection of CBFs (Dowswell 2016); (b) initial imperfection angle of the insert zone for BRBs (Takeuchi et al. 2014).

𝜃𝑖 =𝛿𝑚𝐿0+2𝛿𝑔

𝐿𝑔=

1

500+2 × 1

100= 2.2% (3)

Where 𝛿𝑚 𝐿0⁄ = plumbing tolerance of the brace, 𝛿𝑔 𝐿𝑔⁄ = out-of-flatness tolerance of the gusset.

If accounted for the simultaneous story out-of-plane drift of 1.2%, the SRSS method can be used:

⇒ 𝜃𝑖 = √(2.2)2 + (1.2)2 = 2.5% (4)

Designing a brace for 2.5% of the member compressive force is a simple stability bracing requirement

that has been used in NZS 3404 and in a number of BRB gusset plate designs to date.

As shown in Figure 3(b), in BRBs the initial imperfection at the insert zone may represent the largest source of imperfection, which is also required to be considered (Takeuchi et al. 2014):

𝜃0 =2𝑆𝑟𝐿𝑖𝑛

(5)

Where 𝜃0 = initial imperfection angle of the insert zone, 𝑆𝑟 = clearance between core and restrainer,

𝐿𝑖𝑛 = insert zone length.

Then, the total initial imperfection in BRBs can be approximated by the following equations:

{

𝜃𝑖 =

𝛿𝑚𝐿0+2𝛿𝑔

𝐿𝑔+ 𝜃0

𝜃𝑖 =𝛿𝑚𝐿0+𝛿𝑔

𝐿𝑔+ 𝜃0

gusset plate buckling mode under the yield-line (6)

plastic failure mode over the yield-line (7)

2.2 Yield-line patterns

Based on experimental observations, three different patterns of anticipated yield-line are defined. When

the brace moves out-of-plane without a plastic hinge at the mid-length of it or at the end of the restrainer

in BRBs, the gusset will buckle under the yield-line that runs from the middle of the gusset plate edge

to the underside of the brace and back to the middle of the other gusset plate edge (Figure 4(a)). When

the brace buckles in out-of-plane direction and forms a plastic hinge at the mid-length of it or at the end

of the restrainer in BRBs, plastic failure over the yield-line will occur and the yield-line will run from

the corner of the gusset to the underside of the brace and back to the other corner as shown in Figure

4(b). If the brace is not pulled in close to the beam or column, the yield-line will run from the line of

restraint that occurs at the first re-entrant corner of the gusset to the gusset plate edge in a straight line

and parallel to the end of the brace (Figure 4(c)).

Figure 4. (a) Gusset buckling under the yield-line (Yam & Cheng 2002); (b) plastic failure over the yield-line (interrupted yield-line pattern) (Tsai et al. 2008); (c) plastic failure over the yield-line

(uninterrupted yield-line pattern) (Astaneh-Asl et al. 2006).

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2.3 NLYL Model for CBFs

In CBF connections the system is modelled as a bending element with plastic hinges at the gusset plate.

If the bending moment reaches the reduced plastic moment capacity of the gusset including the axial

force effect, any further increase in load will cause collapse (Dowswell 2016). According to the simple

and very satisfactory plastic analysis, the virtual work method, the stability condition of CBFs is

expressed as follows (Figure 5(a)):

Figure 5. (a) Simplified collapse mechanisms of CBFs; (b) geometry to define the average length,

𝐿𝑎𝑣𝑒; (c) total yield-line length, 𝑏𝑔𝑓; (d) stress distribution within gusset plate for bolted and welded

connections (Astaneh-Asl 1998).

𝑁𝐿𝑎𝑣𝑒𝛿𝑆𝜃 = 2𝑀𝑝𝑔𝜃 ⇒ 𝑀𝑝

𝑔≥𝑁𝐿𝑎𝑣𝑒𝛿𝑆

2= 𝑀𝑦

⋆ (8)

Using an L-shaped model, Dowswell (2006) showed that the out-of-plane restraint is provided by the

portion of the gusset plate with the shorter length of the two distances from brace end to the support𝑠(𝑐). Considering this observation, the plate length in bending is defined in Figure 5(b) and:

𝐿𝑎𝑣𝑒 = 𝑚𝑖𝑛 {(𝑎 + 𝑐) 2⁄

(𝑎 + 𝑏 + 𝑐)/3

(9)

𝛿𝑆 =1

1 −𝑁⋆

𝑁𝑒

≥ 1 (10)

𝑁𝑒 = 𝑓𝑒𝑡𝑔𝑏𝑔𝑎 (11)

𝑓𝑒 =𝜋2𝐸

(𝐿𝑎𝑣𝑒𝑟𝑔)2

(12)

𝐼𝑁:

{

(

𝑁⋆

𝜙𝑁𝑠𝑔)

2

+𝑀𝑦⋆

𝜙𝑀𝑠𝑦𝑔 ≤ 1

(𝑁⋆

𝜙𝑁𝑠𝑔)

2

+𝑀𝑦⋆

𝜙(1.19)𝑀𝑠𝑦𝑔 ≤ 1

regular gusset (13)

stiffened gusset (14)

6

𝑁𝑠𝑔= 𝑡𝑔𝑏𝑔𝑎𝑓𝑦 (15)

𝑀𝑠𝑦𝑔= 𝑆𝑔𝑓𝑦 =

𝑏𝑔𝑓𝑡𝑔2

4𝑓𝑦 (16)

𝑏𝑔𝑓 = 𝑏1 + 𝑏2 + 𝑏3 (17)

Where 𝛿𝑆 = moment amplification factor to account for second-order effect, 𝑁𝑒 = elastic buckling

capacity of gusset, 𝑡𝑔 = gusset thickness, 𝑏𝑔𝑎 = effective width in compression as shown in Figure 5(d)

(dispersion angle is 40°for corner gusset plates and 30°for other configurations), 𝐸 = Young’s modulus

of elasticity, 𝑟𝑔 = radius of gyration of gusset, 𝐼𝑁 = interaction equation from plastic theory for

rectangular and weak-axis bending of wide-flange sections respectively, 𝑀𝑠𝑦𝑔= plastic moment capacity

of gusset, 𝑁𝑠𝑔= nominal section axial capacity of gusset, 𝑆𝑔 = plastic section modulus of gusset, 𝑓𝑦 =

yield stress of the gusset, 𝑏𝑔𝑓 = total length of the yield-line (Figure 5(c)).

2.4 NLYL Model for BRBs

In BRB connections the plastic hinges are assumed at both the gusset plate and restrainer end (Takeuchi

et al. 2014). Given the fact that at least three hinges are required for the system collapse, the stability

conditions of BRBs encompass the following forms (Figure 6):

Figure 6. Collapse mechanisms of BRBs for plastic failure over the yield-line mode: (a) symmetrical mode; (b) one sided mode; (c) asymmetrical mode.

Symmetrical mode:

𝑁𝜉𝐿0𝛿𝑆𝜃 = 𝑀𝑝𝑔𝜃 + 𝑀𝑝

𝑟𝜃 (18)

𝑁𝜉𝐿0𝛿𝑆 ≤ 𝑀𝑝𝑔+𝑀𝑝

𝑟 (19)

One sided mode:

𝑁𝜉𝐿0𝛿𝑆(𝜃1 + 𝜃2) = 𝑀𝑝𝑔𝜃1 +𝑀𝑝

𝑟(𝜃1 + 2𝜃2) (20)

𝑁𝜉𝐿0𝛿𝑆(1 − 𝜉) ≤ (1 − 2𝜉) 𝑀𝑝𝑔+𝑀𝑝

𝑟 (21)

Asymmetrical mode:

𝑁𝜉𝐿0𝛿𝑆(𝜃1 + 𝜃2) = 𝑀𝑝𝑔𝜃1 +𝑀𝑝

𝑟(𝜃1 + 𝜃2) (22)

𝑁𝜉𝐿0𝛿𝑆 ≤ (1 − 2𝜉) 𝑀𝑝𝑔+𝑀𝑝

𝑟 (23)

As can be seen, the asymmetrical mode gives the lowest failure capacity and governs stability.

7

𝛿𝑆 =1

1 −𝑁𝑐𝑢 ⋆

𝑁𝑐𝑟𝐵

≥ 1 (24)

𝑁𝑐𝑟𝐵 =

𝜋2𝐸𝐼𝐵

(𝑘𝐿0)2 ⟶ {

𝑘 = 1

𝑘 = 0.7

𝑁𝑐𝑟𝐵 =

𝜋2𝐸𝐼𝐵(𝑘𝐿0)

2 ⟶ {

𝑘 = 1

𝑘 = 0.7

regular gusset

(25) stiffened gusset

𝑀𝑝𝑔=

{

𝜙𝑀𝑠𝑦

𝑔(1 − (

𝑁𝑐𝑢 ⋆

𝜙𝑁𝑠𝑔)

2

)

1.19𝜙𝑀𝑠𝑦𝑔(1 − (

𝑁𝑐𝑢 ⋆

𝜙𝑁𝑠𝑔)

2

)

regular gusset (26)

stiffened gusset (27)

𝑀𝑝𝑟 = 𝑚𝑖𝑛{ 𝑀𝑝

𝑟−𝑛𝑒𝑐𝑘 , 𝑀𝑝𝑟−𝑟𝑒𝑠𝑡} (28)

𝑀𝑝𝑟−𝑛𝑒𝑐𝑘 =

{

𝜙𝑀𝑠𝑦

𝑛 (1 − (𝑁𝑐𝑢 ⋆ − 𝑁𝑤𝑦

𝑛

𝜙(𝑁𝑦𝑛 − 𝑁𝑤𝑦

𝑛 ))

2

)

𝜙𝑀𝑠𝑦𝑛 (1 − (

𝑁𝑐𝑢 ⋆ − 𝑁𝑤𝑦

𝑛

𝜙(𝑁𝑢𝑛 − 𝑁𝑤𝑦

𝑛 ))

2

)

regular gusset (29)

stiffened gusset (30)

Where 𝜉𝐿0 = BRB connection length from the end of restrainer to the yield-line, 𝐿0 = total BRB length,

𝛿𝑆 = moment amplification factor to account for second-order effect, 𝑁𝑐𝑢⋆ = overstrength compression

capacity of core in BRB, 𝑁𝑐𝑟𝐵 = global elastic buckling capacity of BRB including effect of gusset plate,

𝐸𝐼𝐵 = bending stiffness of restrainer, 𝑀𝑝𝑔= reduced plastic moment capacity of gusset plate including

axial force effect, 𝑀𝑠𝑦𝑔= plastic moment capacity of gusset, 𝑀𝑝

𝑟 = restrainer moment transfer capacity,

𝑀𝑝𝑟−𝑛𝑒𝑐𝑘 = reduced restrainer moment transfer capacity determined by cruciform core plate at the neck

including axial force effect, 𝑀𝑠𝑦𝑛 = plastic moment capacity of neck, 𝑀𝑝

𝑟−𝑟𝑒𝑠𝑡 = restrainer moment

transfer capacity determined by restrainer section at rib end, 𝑁𝑤𝑦𝑛 = yield axial force of cruciform neck

at web zone, 𝑁𝑦𝑛 = yield axial strength of neck, 𝑁𝑢

𝑛 = ultimate axial strength of neck, 𝑆𝑛 = plastic

section modulus at neck.

Takeuchi et al. (2014) also showed that if the insert zone is more than 2.0 times the core plate width, 𝑀𝑝𝑟

is determined by 𝑀𝑝𝑟−𝑛𝑒𝑐𝑘.

In BRBFs there is also a possibility that gusset plate buckles under the yield-line. In this case, the

stability conditions are (Figure 7):

Figure 7. Collapse mechanisms of BRBs for gusset buckling under the yield-line mode: (a) symmetrical mode; (b) asymmetrical mode.

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Symmetrical mode:

𝑁𝐿𝑎𝑣𝑒𝛿𝑆𝜃 = 2𝑀𝑝𝑔𝜃 ⇒ 𝑀𝑝

𝑔≥𝑁𝐿𝑎𝑣𝑒𝛿𝑆

2= 𝑀𝑦

⋆ (31)

Asymmetrical mode:

𝑁𝐿𝑎𝑣𝑒𝛿𝑆 = (2 − 2𝜂) 𝑀𝑝𝑔

⇒ 𝑀𝑝𝑔≥𝑁𝐿𝑎𝑣𝑒𝛿𝑆(2 − 2𝜂)

= 𝑀𝑦⋆ (32)

Similarly, the asymmetrical mode gives the lowest failure capacity and governs stability.

3 CHEVRON CONFIGURATION

In a BRBF with chevron configuration, the flexibility of the frame beam and the bending capacity of

restrainer end considerably affect the stability of the system (Takeuchi et al. 2016). Hikino et al. (2013)

showed that when the insert zone exceeded 1.5 times the width of the core and there was enough moment

transfer capacity at the end of restrainer, the BRB exhibited excellent ductility during the shake table

tests. Otherwise, to prevent the rotation of the beam in out-of-plane direction the recommended stiffness

of the torsional bracing is (Figure 8):

𝐾𝑅 ≥ 2𝑁𝑐𝑢⋆𝐿1(𝐿1 + 𝐿2)

𝐿2 (33)

If sufficient torsional stiffness is not provided by the beam, it is important that adequate lateral bracing

at the midspan of the beam be used. In this case, the BRB connection can be designed using the NLYL

method based on the assumption of a single diagonal configuration.

Figure 8. Geometry of chevron configuration (Hikino et al. 2013).

4 COMPARISON TO TEST DATA

4.1 CBF Specimens

Table 1 presents two full-scale CBF test results (Gross 1990) by which the suitability of different

methods can be assessed. It can be seen that both the AISC and NZS 3404 methods are too conservative

and overdesign the gusset plate. Several past experiments (Lopez et al. 2002; Uriz & Mahin 2008;

Palmer et al. 2016) have shown that in-plane stiffness of the gusset plate can have detrimental effects on

the seismic performance of a steel braced frame. In a number of these experiences, column and beam

hinging or fracture was observed as a result of the generation of a large in-plane bending moment when

story undergoes large inelastic drifts. Designing a gusset plate by means of the NLYL method will yield

the minimum required thickness and reduce the mobilization of this in-plane stiffness.

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Table 1. Comparison of predicted failure axial force in different methods to the physical CBF tests

Specimen

Failure axial force (kN)

AISC

(𝑘 = 0.5, 𝑙𝑏 = 𝑙1) NZS 3404

(𝑘 = 0.7, 𝑙𝑏 = 𝑙1) NLYL

Experiment (Gross 1990)

General yield

capacity

Ultimate buckling

capacity

No. 1 319 221 380 396 516

No. 2 319 221 380 400 614

4.2 BRB Specimens

Comparison between different methods and the results from six BRB tests (Takeuchi et al. 2014) is

summarized in Table 2. It is observed that the NLYL method produces suitably conservative results,

however, neither the AISC nor NZS 3404 can predict the failure axial force conservatively owing to a

larger initial imperfection in BRB specimens.

On the other hand, although theoretically the asymmetrical mode governs stability, in some cases it is

also possible that a BRB buckles in a mode with higher compressive capacity since the final buckling

mode would primarily depend on the initial imperfection shape. This could be one of the sources of

conservatism in both the Takeuchi and NLYL methods.

Table 2. Comparison of predicted failure axial force in different methods to the physical BRB tests

Specimen

Failure axial force (kN)

Gusset buckling under the yield-line Plastic failure over the yield-line Experiment

(Takeuchi 2014) AISC

(𝑘 = 0.5; 𝑙𝑏 = 𝑙1) NZS 3404

(𝑘 = 0.7; 𝑙𝑏 = 𝑙1) NLYLgb NLYLpf Takeuchi method

MRL1.0S1H 620 561 778 725 818 -

MRL2.0S1 620 561 608 436 520 535

MRL2.0S2 620 561 556 367 410 507

MCL2.0S2 620 561 562 385 432 375

MRL1.0S1 620 561 565 326 345 362

MRL1.0S2 620 561 482 207 217 300

* MRL1.0S1H did not fail in experiment

5 CONCLUSIONS

A simplified stability design method is proposed in this paper, based on the concept of notional load

method, which aims to determine the minimum size of gusset plates necessary to provide the out-of-

plane stability condition in actual design. A comparison between the computed results obtained from the

NLYL method and the available testing shows that it is suitably conservative and can be used for

different gusset plate configurations in both CBFs and BRBFs.

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REFERENCES

AISC. (2012). Seismic design manual (2nd ed.). Chicago, IL: American Institute of Steel Construction.

Astaneh-Asl, A. (1998). Seismic behaviour and design of gusset plates. (Steel Technical Information and Product Services (Steel TIPS)), Moraga, CA: Structural Steel Educational Council.

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11

Example 1. NLYL Method for CBF Specimen No. 1 (Gross, 1990)

Figure 9. Specimen No. 1 (Gross 1990)

𝐹𝑦 = 46.7 𝑘𝑠𝑖 (322 𝑀𝑃𝑎)

𝑡𝑔 = 0.25 𝑖𝑛 (6.35 𝑚𝑚)

𝑁 = 𝜃𝑖𝑁⋆ = 0.022 𝑁⋆

𝑏𝑔𝑎 = [2× 𝑡𝑎𝑛(40°) × 4] + 3 = 9.71 𝑖𝑛 (247 𝑚𝑚)

𝐴𝑔 = 𝑡𝑔𝑏𝑔𝑎 = 0.25 × 9.71 = 2.43 𝑖𝑛2 (1566 𝑚𝑚2)

𝐿𝑎𝑣𝑒 = 𝑚𝑖𝑛 [(𝑎 + 𝑐) 2⁄ = (7 + 1) 2⁄ = 4 𝑖𝑛 (101.6 𝑚𝑚)

(𝑎 + 𝑏 + 𝑐)/3 = (7 + 12.5 + 1)/3 = 6.83 𝑖𝑛 (173.5 𝑚𝑚)

𝑏𝑔𝑓 = 𝑏1 + 𝑏2 = 7.5 + 11.6 = 19.1 𝑖𝑛 (485 𝑚𝑚)

𝑓𝑒 =𝜋2𝐸

(𝐿𝑎𝑣𝑒𝑟𝑔)2 =

𝜋2 × 2 × 105

(101.61.833

)2 = 642 𝑀𝑝𝑎

𝑁𝑒 = 𝑓𝑒𝐴𝑔 = 642 × 1566 = 1005 𝑘𝑁

𝛿𝑆 =1

1 −𝑁⋆

𝑁𝑒

=1

1 −𝑁⋆

1005

𝑀𝑦⋆ =

𝑁𝐿𝑎𝑣𝑒𝛿𝑆2

=0.022 × 𝑁⋆ × 101.6

2×

1

1 −𝑁⋆

1005

𝑆𝑔 =𝑏𝑔𝑓 𝑡𝑔

2

4=485 × 6.352

4= 4889 𝑚𝑚3

(𝑁⋆

𝑁𝑠𝑔)

2

+𝑀𝑦⋆

𝑀𝑠𝑦𝑔 ≤ 1 ⇒ (

𝑁⋆

1566 × 322)

2

+0.022 × 𝑁⋆ × 101.6

2 × (1 −𝑁⋆

1005 × 103) × 322 × 4889

≤ 1

→ 𝑁⋆𝑚𝑎𝑥 = 380 𝑘𝑁

12

Example 2. NLYL Method for BRBs Specimen MRL2.0S1 (Takeuchi et al. 2014)

Figure 10. Specimen MRL2.0S1 (Takeuchi et al. 2014)

First assume plastic hinge location at the neck:

𝑀𝑝𝑔= 𝑀𝑠𝑦

𝑔(1 − (

𝑁𝑐𝑢⋆

𝑁𝑠𝑔)

2

) 𝑀𝑝𝑟−𝑛𝑒𝑐𝑘 = 𝑀𝑠𝑦

𝑛 (1 − (𝑁𝑐𝑢⋆ − 𝑁𝑤𝑦

𝑛

𝑁𝑦𝑛 − 𝑁𝑤𝑦

𝑛)

2

)

𝜃𝑖 =𝑎𝑟

𝜉𝐿0=6.8

416= 0.0163 𝑁𝑤𝑦

𝑛 = (𝐵𝑐 − 𝑡𝑅)(𝑡𝑐)(𝑓𝑦) = (78)(12)(266.8) = 250 𝑘𝑁

𝑁𝑦𝑛 = [𝐵𝑐𝑡𝑐 + (𝐵𝑅 − 𝑡𝑐)(𝑡𝑅)] × 𝑓𝑦 = [(90)(12) + (90 − 12)(12)] × 266.8 = 537.9 𝑘𝑁

𝑁𝑠𝑔= 𝑡𝑔𝑏𝑔𝑎𝑓𝑦 =

(12)(240)(266.8)

1000= 768.4 𝑘𝑁

𝑀𝑠𝑦𝑔= 𝑆𝑔𝑓𝑦 =

𝑏𝑔𝑓𝑡𝑔2

4𝑓𝑦 =

(240)(12)2

4(266.8)=2.305 kNm

𝑀𝑠𝑦𝑛 = 𝑆𝑛𝑓𝑦 = (

(12)(90)2

4+(78)(12)2

4) (266.8)(10)−6 = 7.23 𝑘𝑁𝑚

𝑁𝑐𝑟𝐵 =

𝜋2𝐸𝐼𝐵(𝑘𝐿0)

2=(𝜋)2(5.81)(10)11

(1 × 2392)2= 1002 𝑘𝑁

𝑁𝜉𝐿0𝛿𝑆 ≤ (1 − 2𝜉) 𝑀𝑝𝑔+𝑀𝑝

𝑟−𝑛𝑒𝑐𝑘

(0.0163)(𝑁𝑐𝑢⋆ )(0.416)(

1

1 −𝑁𝑐𝑢⋆

1002

) ≤ (0.652)(1 − (𝑁𝑐𝑢⋆

768.4)

2

) (2.305) + (1 − (𝑁𝑐𝑢⋆ − 250

537.9 − 250)

2

) (7.23)

⟹ 𝑁𝑐𝑢 (𝑚𝑎𝑥)⋆ ≤ 436 𝑘𝑁 ⟶𝑀𝑝

𝑟−𝑛𝑒𝑐𝑘 = 4.21 𝑘𝑁𝑚

Assume plastic hinge location at the restrainer end (Takeuchi et al. 2014):

𝑀𝑝𝑟−𝑟𝑒𝑠𝑡 = 𝑚𝑖𝑛{𝑆𝑟𝑝𝜎𝑟𝑦; 𝛼𝑝

𝑟[𝐾𝑅𝑟1𝜃𝑦1′ + 𝐾𝑅2(𝜃𝑦2 − 𝜃𝑦1

′ )]} 𝑆𝑟𝑝𝜎𝑟𝑦 = (51947)(385.8)

= 20.04 𝑘𝑁𝑚

𝛼𝑝𝑟 = 4.5 − 1.5(𝐿𝑖𝑛 𝐵𝑐) =⁄ 4.5 − 1.5(180 90) = 1.5⁄

𝑎 =𝐵𝑟 − 𝐵𝑐

4=125 − 90

4= 8.75 𝑚𝑚

𝐾𝑅𝑟1 =𝐸𝐵𝑟𝑡𝑟

3𝐿𝑖𝑛3

3(2𝐵𝑟𝑎3 − 3𝑎4)

=(2.05 × 105)(125)(2.3)3 (180)3

3(2 × 125 × 8.753 − 3 × 8.754)= 4043490617

𝜃𝑦1′ = 1.64 × 10−3(𝜎𝑟𝑦 𝐸⁄ )(𝐵𝑟 𝑡𝑟⁄ )(𝐵𝑐 𝐿𝑖𝑛⁄ )

= 1.64 × 10−3(385.8 (2.05 × 105)⁄ )(125 2.3⁄ )(90 180⁄ ) = 8.387 × 10−5

𝐾𝑅2 = 0.11𝜎𝑟𝑦𝐵𝑟3(𝐿𝑖𝑛 𝐵𝑐⁄ )3 = 0.11(385.8)(125)3(180 90⁄ )3 = 663093750

𝜃𝑦2 =𝐵𝑟

𝐿𝑖𝑛√(𝜎𝑟𝑦

2𝐸)2

+ (𝑎𝜎𝑟𝑦

𝐵𝑟𝐸) =

125

180√(

385.8

2(2.05 × 105))2

+ (8.75(385.8)

125(2.05 × 105)) = 7.997 × 10−3

⟹𝑀𝑝𝑟−𝑟𝑒𝑠𝑡 = 8.38 𝑘𝑁𝑚 ⟶ 𝑁𝑐𝑢 (𝑚𝑎𝑥)

⋆ ≤ 572 𝑘𝑁